Lemma 2.1 ([1]};[2]};[3]})
Let \(2<p<3\)  and \(\mu ,a,\alpha >0\)  in (REF ), there exists \(\tau (\alpha ,p)>0\)  such that if \(\mu a^{p-p\gamma _p}\le \tau (\alpha ,p)\)  then \( m^{\pm }_{p,\mu ,\alpha }(a)=\inf \limits _{u\in \mathcal {T}^{\pm }_{a,p,\mu ,\alpha } }\mathcal {A}_{p,\mu ,\alpha }(u)<0\)  and it can be attained by some \(u^{\pm }_{a,\mu ,\alpha }\)  which is real valued, positive, radially symmetric and radially decreasing. Moreover, \(m^{\pm }_{p,\mu ,\alpha }(a)\)  is strictly decreasing with respect to \(a>0\) .

